中文摘要：

同宿轨的求解是非线性系统领域的核心问题之一,特别是对动力系统分岔与混沌的研究有重要意义.根据同宿轨的几何特点,采用轨线逼近的方式,通过定义逼近轨线与鞍点的距离,将同宿轨的求解转化为求距离最小值的无约束非线性优化问题.为了提高优化结果的完整性,还提出了基于区间细分的搜索算法和实现方法,并找出了Lorenz系统,Shimizu-Morioka系统和超混沌Lorenz系统等的多个同宿轨道和对应参数,验证了本文方法的有效性.

英文摘要：

Detecting homoclinic orbits is a key problem in nonlinear dynamical systems,especially in the study of bifurcation and chaos.In this paper,we propose a new method to solve the problem with trajectory optimization.By defining a distance between a saddle point and its near trajectories,the problem becomes a common problem in unconstrained nonlinear optimization to minimize the distance.A subdivision algorithm is also proposed in this paper to improve the integrity of results.By applying the algorithm to the Lorenz system,the Shimizu-Morioka system and the hyperchaotic Lorenz system,we successfully find many homoclinic orbits with the corresponding parameters,which suggests that the method is effective.